An efficient wavelet-based approximation method for the coupled nonlinear fractal–fractional 2D Schrödinger equations

In this study, a fast semi-discrete approach based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of a new class of the coupled nonlinear time fractal–fractional Schrodinger equations. Although the proposed method can be applied for any fractional derivative, we focus on the Atangana–Reimann–Liouville derivative with Mittag–Leffler kernel due to its privileges. To carry out the method, first the fractal–fractional derivatives need to be discretized through the finite difference scheme and the \(\theta \)-weighted method to derive a recurrent algorithm. Then, the unknown solution of the intended problem is expanded in terms of the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the problem becomes reduced to a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the validity and accuracy of this practicable wavelet method, some numerical test examples are provided. The achieved numerical results manifest the exponential accuracy of approximate solutions of the new fractal–fractional model.